This Metric-Affine framework has received a renewed attention over the last years [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [14]}, [15]} especially in regards to its cosmological applications [16]}, [17]}, [18]}, [19]}, [20]}, [21]}, [22]}, [23]}, [24]}, [25]}, [26]}. It therefore becomes apparent that it is of great importance to have a complete setup for both the geometrical and the matter structures on this non-Riemannian Cosmology. Regarding the latter point, we had earlier developed in [17]} the generalized fluid notion that sources the non-Riemannian degrees of freedom in a Cosmological setup. We named this perfect fluid generalization (where the microscopic characteristics of matter are taken into account) the Perfect Cosmological Hyperfluid [17]}Quite recently we have generalized this hyperfluid notion even further, even outside of the realm of Cosmology [29]}.. There it was shown that this kind of fluid is the appropriate one for the description of the matter sources in Metric-Affine Cosmology, since it sources all the non-Riemannian degrees of freedom that are allowed by the Cosmological Principle. Having the complete set for the sources one also needs the associated compatible set of geometrical objects in this Metric-Affine Cosmological setting. These are torsion, non-metricity and curvature, the three fundamental geometrical objects of the non-Riemannian geometry [30]}. The covariant expressions for both torsion and non-metricity, compatible with the Cosmological Principle, have also been presented in [17]}. Therefore, the only remaining missing link is the covariant expression for the Riemann (curvature) tensor that respects both isotropy and homogeneity. It is this gap that a part of this work here intends to fill. It goes without saying that having the exact covariant form of the Riemann tensor serves as a very helpful tool for studying the Cosmological aspects of Metric-Affine Gravity and especially Theories that are constructed by the latter, such as \(f(R)\)  [5]}, [33]} or certain generalizations of it [34]}.
Of course, having the covariant form of the Riemann tensor one could then go on and compute several important by-products of the latter such as Ricci tensor, Ricci scalar, Riemann dual etc. We list several of these quantities after the derivation of the Riemann tensor. In addition, one highly interesting quantity one can construct, when having this covariant form of the Riemann tensor, is the well-known Gauss-Bonnet density. Using a certain generalization of the aforementioned density for non-Riemannian spaces, we derive its form for a homogeneous torsionful and non-metric Cosmology. As expected we find that when the full non-metricity is present the Gauss-Bonnet density fails to be an exact differential [35]}, [36]}. It is also known that for the restricted class of Cartan-Weyl spaces the generalized Gauss-Bonnet density is indeed topological. In our Cosmological setting we extend this result here for a slightly more general class of non-metricity, that is for geometries that generalize the Cartan-Weyl structure.
